IB Mathematics Analysis and Approaches Topic SL 5.3—differentiating Polynomials, n E Z Questionbank (SL/HL)

IB Mathematics Analysis and Approaches Topic SL 5.3—differentiating Polynomials, n E Z Questionbank (SL/HL) | RevisionDojo

Practice IB Mathematics Analysis and Approaches (AA) Topic SL 5.3—differentiating Polynomials, n E Z with authentic exam-style questions for both SL and HL students. This question bank focuses on the exact syllabus content for SL 5.3—differentiating Polynomials, n E Z and mirrors Paper 1, 2, 3 style where relevant.

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Paper

Level

Question 1

SLPaper 1

The function $f (x) = (3 x - 2) (x^{2} + 4 x + 5)$ .

Expand $f (x)$ .

[2]

Find $f^{'} (x)$ .

[3]

Determine the equation of the tangent to $f$ at $x = 1$ .

[3]

Find the coordinates of the points where the tangent is horizontal.

[4]

Question 2

SLPaper 1

Consider the function $f (x) = x^{5} - 2 a x^{3} + b x^{2} + 4$ , where $a$ and $b$ are constants. The graph of $y = f (x)$ has a horizontal tangent at $x = 1$ , and the point $(1, 7)$ lies on the graph.

Show that $a = \frac{9}{2}$ and $b = 11$ .

[5]

Find the second derivative $f^{''} (x)$ .

[3]

Find the x-coordinates of any stationary points of $f (x)$ and determine their nature.

[3]

Find the intervals where $f (x)$ is increasing.

[3]

Determine the number of solutions to $f (x) = 7$ in the interval $[- 2, 3]$ .

[3]

Question 3

HLPaper 2

Consider the function $f (x) = \frac{a}{x ^{2} + 1} + b x^{2}$ , where $a, b > 0$ and $x \in R$ . The graph of $y = f (x)$ has a horizontal tangent at $x = 1$ , and passes through the point $P (1, 1.5)$ .

Find $f^{'} (x)$ .

[3]

Show that $a = 2$ and $b = \frac{1}{2}$ .

[5]

Show that $f (x)$ has a local minimum at $x = 1$ .

[3]

Find the exact area under the curve from $x = 0$ to $x = 1$ .

[4]

The curve lies above the $x$ -axis for $0 \leq x \leq 2$ . The region $R$ is bounded by the graph of $f$ , the $x$ -axis, and the lines $x = 0$ and $x = 2$ . Find the volume of the solid formed by rotating $R$ about the $x$ -axis.

[5]

Question 4

SL & HLPaper 1

Consider the function $f (x) = sin (x) + cos (x)$ , where $x$ is in radians.

Differentiate $f (x)$ .

[2]

Determine the critical points of $f (x)$ in the interval $[0, 2 π]$ .

[3]

Classify the critical points found in Part 2 as local maxima, minima, or inflection points.

[4]

Question 5

SLPaper 1

Consider a cylinder of radius $5 r$ and height $h$ . A smaller cylinder of radius $r$ is removed from the centre to form a hollow cylinder. This is shown in the following diagram. All lengths are measured in centimetres. The total surface area of the hollow cylinder, in cm $^{2}$ , is given by $S$ . The volume of the hollow cylinder, in cm $^{3}$ , is given by $V$ .

Show that $S = 48 π r^{2} + 12 π r h$ .

[3]

The total surface area of the hollow cylinder is $480 π$ cm $^{2}$ . Show that $V = 960 π r - 96 π r^{3}$ .

[6]

Find an expression for $\frac{d V}{d r}$ .

[2]

The hollow cylinder has its maximum volume when $r = p \frac{5}{6}$ , where $p \in Z^{+}$ . Find the value of $p$ .

[3]

Hence, find this maximum volume, giving your answer in the form $q π \frac{5}{6}$ , where $q \in Z^{+}$ .

[3]

IB Mathematics Analysis and Approaches SL 5.3—differentiating Polynomials, n E Z Questionbank

Unlock all 18 questions with a free account

IB Mathematics Analysis and Approaches SL 5.3—differentiating Polynomials, n E Z Questionbank

IB Mathematics Analysis and Approaches (AA) Topic SL 5.3—differentiating Polynomials, n E Z Questionbank

Unlock all 18 questions with a free account